# Math Help - complex bilinear extension

1. ## complex bilinear extension

Hello,

I came across the following statement:

"..., where g is the complex bilinear extension of the standard scalar product on R^n to C^n (note that this a symmetric form, not a Hermitian one)."

Can someone tell me how the "complex bilinear extension" is defined, i.e. if x,y in C^n, then g(x,y)=... .

The "standard" scalar product on R^n is $\sqrt{xy}$ where x and y are vectors in R^n. If x and y are vectors in C^n, the scalar product is $\sqrt{xy^*}$ where "y*" indicates the complex conjugate of y: if $y= (a_1+ ib_1, a_2+ ib_2, ..., a_n+ ib_n)$ then $y^*= (a_1- ib_1, a_2- ib_2, ..., a_n- ib_n)$. That is, if $x= (c_1+ id_1, c_2+ id_2, ..., c_n+ id_n)$ and $y= (a_1+ ib_1, a_2+ ib_2, ..., a_n+ ib_n)$ then the inner product of x and y is $(c_1+ id_1)(a_1- ib_1)+ (c_2+ id_2)(a_2- ib_2)+ ...+ (c_n+ id_n)(a_1- ib_n)$.